Journal of Lie Theory 20 (2010), No. 3, 543--580
Copyright Heldermann Verlag 2010
Global Lie Symmetries of the Heat and Schrödinger Equation
Mark R. Sepanski
Dept. of Mathematics, Baylor University, One Bear Place 97328, Waco, TX 76798-7328, U.S.A.
Mark_Sepanski@baylor.edu
Ronald J. Stanke
Dept. of Mathematics, Baylor University, One Bear Place 97328, Waco, TX 76798-7328, U.S.A.
Ronald_Stanke@baylor.edu
We examine solutions to a family of differential equations, including the heat and Schrödinger equations, that are globally invariant under the action of the corresponding Lie symmetry group. The solution space is realized in a nonstandard parabolically induced representation space as the kernel of a linear combination of Casimir operators of certain distinguished subgroups. Composition series provide a complete description of this kernel and, for special inducing parameters, the oscillator representation is realized in a natural and explicit way as a subspace of solutions to the Schrödinger equation.
Keywords: Heat equation, Schroedinger equation, oscillator representation.
MSC: 58J70, 22E45, 22E70, 35A30
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